Quasi-isometry

The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.

The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.

The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most

In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance

of it, so rounding changes the distance between pairs of points by adding or subtracting at most

Every pair of finite or bounded metric spaces is quasi-isometric.

Given a finite generating set S of a finitely generated group G, we can form the corresponding Cayley graph of S and G. This graph becomes a metric space if we declare the length of each edge to be 1.

[3] This quasi-isometry class is thus an invariant of the group G. Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods.

More generally, the Švarc–Milnor lemma states that if a group G acts properly discontinuously with compact quotient on a proper geodesic space X then G is quasi-isometric to X (meaning that any Cayley graph for G is).

This gives new examples of groups quasi-isometric to each other: A quasi-geodesic in a metric space

The fact that in some spaces the converse is coarsely true, i.e. that every quasi-geodesic stays within bounded distance of a true geodesic, is called the Morse Lemma (not to be confused with the Morse lemma in differential topology).

Formally the statement is: It is an important tool in geometric group theory.

An immediate application is that any quasi-isometry between proper hyperbolic spaces induces a homeomorphism between their boundaries.

This result is the first step in the proof of the Mostow rigidity theorem.

Furthermore, this result has found utility in analyzing user interaction design in applications similar to Google Maps.

[5] The following are some examples of properties of group Cayley graphs that are invariant under quasi-isometry:[2] A group is called hyperbolic if one of its Cayley graphs is a δ-hyperbolic space for some δ.

:[6] indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words.

Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n. According to Gromov's theorem, a group of polynomial growth is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index.

grows more slowly than any exponential function, G has a subexponential growth rate.

That is, each end represents a topologically distinct way to move to infinity within the space.

The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is independent of the choice of a finite generating set.

If two connected locally finite graphs are quasi-isometric then they have the same number of ends.

[7] In particular, two quasi-isometric finitely generated groups have the same number of ends.

The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox.

In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.

In this setting, a group is amenable if one can say what proportion of G any given subset takes up.

An important class of ultralimits are the so-called asymptotic cones of metric spaces.

One often takes the base-point sequence to be constant, pn = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by

The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.

[9] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.